Handling the measurement error problem by means of panel data: Moment methods applied on firm data



etc. Here X i() denotes the [(T - 2) × K] matrix of x levels obtained by deleting
rows t and θ from Xi., and ∆Xi(t) denotes the [(T - 2) × K] matrix of x differences
obtained by stacking all one-period differences between rows of
Xi. not including
period
t and the single two-period difference between the columns for periods t +1
and
t 1. The vectors yi() and ∆yi(t) are constructed from yi in a similar way.
In general, we let subscripts (
) and (t) on a matrix or vector denote deletion of
(
) differences and t levels, respectively. Stacking yi0(tθ) , ∆yi0(t) , xi() , and ∆xi(t)
by individuals, we get

y1( )

δ y01( t )

x1(tθ)

δ x 1( t )

Y(tθ)=

.

.

.

, δ Y ( t ) =

.

.

.

, x ( tθ ) =

.

.

.

,X ( t ) =

.

.

.

y0N (tθ)

δ yN ( t )

xN (tθ)

xN (t)

which have dimensions (N × (T 2)), (N × (T 2)), (N × (T 2)K), and (N ×
(T 2)K), respectively. These four matrices contain the alternative IV sets in the
GMM procedures to be considered below.

Equation in differences, IV’s in levels. Using X () as IV matrix for ∆X ,we
obtain the following estimator of
β, specific to period (t, θ) differences and utilizing
all admissible
x level IV’s,

(38)


βDx(tθ) = (X) 0X (tθ) X (0tθ)X (tθ)


1 X(0tθ) (X)


-1


(X) 0X (tθ)  X (0tθ) X (tθ)


X (0tθ) (y )


= hPi(∆xitθ) 0x


i(tθ)ihPi xi0(tθ) xi(tθ) i    hPi xi0(tθ) (∆xitθ


-1


hPi (∆xitθ) xi(tθ) ihPi xi(tθ) xi(tθ) i   hPi xi

It exists if X (0tθ)X (tθ) has rank (T 2)K, which requires N (T 2)K. This
estimator examplifies (23), utilizes the OC
E[x0()(∆eitθ)] = 0(T-2)K, 1 - which
follows from (32) - and minimizes the quadratic form

[N-1X(0tθ)]0[N-2X(0tθ)X(tθ)]-1[N-1X(0tθ)].

18



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